Probability distributions for discrete Fourier spectra

Abstract

The paper analyses power-spectrum estimates obtained by fast-Fourier-transform techniques. Distributions are obtained for data, augmented, where necessary, by sequences of zeros, and the effect of data smoothing on the reliability of the estimates is considered. The effect of segment averaging is analysed and a joint probability distribution is derived for the resulting spectrum estimates. The number of degrees of freedom per estimate can then be directly determined. 1st- and 2nd-order moments of logarithmic spectra are derived which lead to confidence bands on the spectral estimates. Frequency-domain smoothing is then considered, and it is shown, that, for specified lengths of Gaussian random data, this, unlike data smoothing, does not lead to a reduction in the number of degrees of freedom. Finally, the general case of frequency smoothing followed by adjacent estimate averaging is analysed. A factor is proposed for assessing loss of stability of such estimates. Computer results are given which demonstrate the effects of several data windows and sets of frequency-smoothing coefficients. Results in the appendixes show that loss in degrees of freedom is related to the eigenvalues of a specific covariance matrix.